Cubes and Squares

Let the two numbers be $x$ and $y$ . Then $x + y = 3$ and

$x^{3} + y^{3} = 25 \Longrightarrow (x + y)(x^{2} - xy + y^{2}) = 25 \Longrightarrow x^{2} + y^{2} = \dfrac{25}{3} + xy,$ (A).

Now $(x + y)^{2} = 9 \Longrightarrow x^{2} + y^{2} + 2xy = 9 \Longrightarrow x^{2} + y^{2} = 9 - 2xy,$ (B).

Multiplying equation (A) through by $2$ and adding the result to (B) gives us that

$3(x^{2} + y^{2}) = \dfrac{50}{3} + 9 = \dfrac{77}{3} \Longrightarrow x^{2} + y^{2} = \boxed{\dfrac{77}{9}}.$